Riemann invariant

Consider the set of conservation equations:

:$l\_i\backslash left(A\_\{ij\}\; \backslash frac\{\backslash partial\; u\_j\}\{\backslash partial\; t\}\; +a\_\{ij\}\backslash frac\{\backslash partial\; u\_j\}\{\backslash partial\; x\}\; \backslash right)+l\_j\; b\_j=0$

where $A\_\{ij\}$ and $a\_\{ij\}$ are the elements of the matrices $\backslash mathbf\{A\}$ and $\backslash mathbf\{a\}$ where $l\_\{i\}$ and $b\_\{i\}$ are elements of vectors. It will be asked if it is possible to rewrite this equation to

:$m\_j\backslash left(\backslash beta\backslash frac\{\backslash partial\; u\_j\}\{\backslash partial\; t\}\; +\backslash alpha\backslash frac\{\backslash partial\; u\_j\}\{\backslash partial\; x\}\; \backslash right)+l\_j\; b\_j=0$

To do this curves will be introduced in the $(x,t)$ plane defined by the vector field $(\backslash alpha,\backslash beta)$. The term in the brackets will be rewritten in terms of a total derivative where $x,t$ are parametrized as $x=X(\backslash eta),t=T(\backslash eta)$

:$\backslash frac\{d\; u\_j\}\{d\; \backslash eta\}=T\text{'}\backslash frac\{\backslash partial\; u\_j\}\{\backslash partial\; t\}+X\text{'}\backslash frac\{\backslash partial\; u\_j\}\{\backslash partial\; x\}$

comparing the last two equations we find

:$\backslash alpha=X\text{'}(\backslash eta),\; \backslash beta=T\text{'}(\backslash eta)$

which can be now written in characteristic form

:$m\_j\backslash frac\{du\_j\; \}\{\; d\; \backslash eta\; \}+l\_jb\_j\; =\; 0$

where we must have the conditions

:$l\_iA\_\{ij\}=m\_jT\text{'}$

:$l\_ia\_\{ij\}=m\_jX\text{'}$

where $m\_j$ can be eliminated to give the necessary condition

:$l\_i(A\_\{ij\}X\text{'}-a\_\{ij\}T\text{'})=0$

so for a nontrivial solution is the determinant

:$|A\_\{ij\}X\text{'}-a\_\{ij\}T\text{'}|=0$

For Riemann invariants we are concerned with the case when the matrix $\backslash mathbf\{A\}$ is an identity matrix to form

:$\backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; t\}\; +a\_\{ij\}\backslash frac\{\backslash partial\; u\_j\}\{\backslash partial\; x\}=0$

notice this is homogeneous due to the vector $\backslash mathbf\{n\}$ being zero. In characteristic form the system is

:$l\_i\backslash frac\{du\_i\; \}\{dt\; \}=0$ with $\backslash frac\{dx\; \}\{dt\; \}=\backslash lambda$

Where $l$ is the left eigenvector of the matrix $\backslash mathbf\{A\}$ and $\backslash lambda\; \text{'}s$ is the characteristic speeds of the eigenvalues of the matrix $\backslash mathbf\{A\}$ which satisfy

:$|A\; -\backslash lambda\backslash delta\_\{ij\}|=0$

To simplify these characteristic equations we can make the transformations such that $\backslash frac\{dr\}\{dt\}=l\_i\backslash frac\{du\_i\}\{dt\}$

which form

:$\backslash mu\; l\_idu\_i\; =dr$

An integrating factor $\backslash mu$ can be multiplied in to help integrate this. So the system now has the characteristic form

:$\backslash frac\{dr\}\{dt\; \}=0$ on $\backslash frac\{dx\}\{dt\}=\backslash lambda\_i$

which is equivalent to the diagonal system

{{cite book

|last=Whitham |first=G. B.

|year=1974

|title=Linear and Nonlinear Waves

|publisher=Wiley

|page=

|isbn=978-0-471-94090-6

}}

:$r\_t^k\; +\backslash lambda\_kr\_x^k=0,$ $k=1,...,N.$

The solution of this system can be given by the generalized hodograph method.

{{cite book

|last=Kamchatnov |first=A. M.

|year=2000

|title=Nonlinear Periodic Waves and their Modulations

|publisher=World Scientific

|page=

|isbn=978-981-02-4407-1

}}{{cite journal

|last=Tsarev

|first=S. P.

|year=1985

|title=On Poisson brackets and one-dimensional hamiltonian systems of hydrodynamic type

|url=http://ikit.institute.sfu-kras.ru/files/ikit/Tsarev-DAN-1985-eng.pdf

|journal=Soviet Mathematics - Doklady

|volume=31

|issue=3

|pages=488–491

|mr=2379468

|zbl=0605.35075

|access-date=2011-08-20

|archive-date=2012-03-30

|archive-url=https://web.archive.org/web/20120330233314/http://ikit.institute.sfu-kras.ru/files/ikit/Tsarev-DAN-1985-eng.pdf

|url-status=dead

}}

Consider the one-dimensional Euler equations written in terms of density $\backslash rho$ and velocity $u$ are

:$\backslash rho\_t+\backslash rho\; u\_x+u\backslash rho\_x=0$

:$u\_t+uu\_x+(c^2/\backslash rho)\backslash rho\_x=0$

with $c$ being the speed of sound is introduced on account of isentropic assumption. Write this system in matrix form

:

where the matrix $\backslash mathbf\{a\}$ from the analysis above the eigenvalues and eigenvectors need to be found. The eigenvalues are found to satisfy

:$\backslash lambda^2-2u\backslash lambda+u^2-c^2=0$

to give

:$\backslash lambda=u\backslash pm\; c$

and the eigenvectors are found to be

:$\backslash left(\; \backslash begin\{matrix\}\; 1\backslash \backslash \; \backslash frac\{c\; \}\{\backslash rho\; \}\; \backslash end\{matrix\}\backslash right),\backslash left(\; \backslash begin\{matrix\}\; 1\backslash \backslash \; -\backslash frac\{c\; \}\{\backslash rho\; \}\; \backslash end\{matrix\}\backslash right)$

where the Riemann invariants are

:$r\_1=J\_+=u+\backslash int\; \backslash frac\{c\}\{\backslash rho\}d\backslash rho,$

:$r\_2=J\_-=u-\backslash int\; \backslash frac\{c\}\{\backslash rho\}d\backslash rho,$

($J\_+$ and $J\_-$ are the widely used notations in gas dynamics). For perfect gas with constant specific heats, there is the relation $c^2=\backslash text\{const\}\backslash ,\; \backslash gamma\; \backslash rho^\{\backslash gamma-1\}$, where $\backslash gamma$ is the specific heat ratio, to give the Riemann invariantsZelʹdovich, I. B., & Raĭzer, I. P. (1966). Physics of shock waves and high-temperature hydrodynamic phenomena (Vol. 1). Academic Press.Courant, R., & Friedrichs, K. O. 1948 Supersonic flow and shock waves. New York: Interscience.

:$J\_+=u+\backslash frac\{2\}\{\backslash gamma-1\}c,$

:$J\_-=u-\backslash frac\{2\}\{\backslash gamma-1\}c,$

to give the equations

:$\backslash frac\{\backslash partial\; J\_+\}\{\backslash partial\; t\}+(u+c)\backslash frac\{\backslash partial\; J\_+\}\{\backslash partial\; x\}=0$

:$\backslash frac\{\backslash partial\; J\_-\}\{\backslash partial\; t\}+(u-c)\backslash frac\{\backslash partial\; J\_-\}\{\backslash partial\; x\}=0$

In other words,

:$$

\begin{align}

&dJ_+ = 0, \, J_+=\text{const}\quad \text{along}\,\, C_+\, :\, \frac{dx}{dt}=u+c, \\

&dJ_- = 0, \, J_-=\text{const}\quad \text{along}\,\, C_-\, :\, \frac{dx}{dt}=u-c,

\end{align}

where $C\_+$ and $C\_-$ are the characteristic curves. This can be solved by the hodograph transformation. In the hodographic plane, if all the characteristics collapses into a single curve, then we obtain simple waves. If the matrix form of the system of pde's is in the form

:$A\backslash frac\{\backslash partial\; v\}\{\backslash partial\; t\}+B\backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\}=0$

Then it may be possible to multiply across by the inverse matrix $A^\{-1\}$ so long as the matrix determinant of $\backslash mathbf\{A\}$ is not zero.

• Simple wave

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{{Bernhard Riemann}}

Category:Invariant theory

Category:Partial differential equations

Category:Conservation equations

Category:Bernhard Riemann